Systems and Methods for Estimating Fluid Breakthrough Times at Producing Well Locations

ABSTRACT

Systems and methods for estimating fluid breakthrough times at producing well locations based on fluid propagation simulation.

CROSS-REFERENCE TO RELATED APPLICATIONS

None

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

Not applicable.

FIELD OF THE INVENTION

The present invention generally relates to estimating fluid breakthrough times at producing well locations. More particularly, the invention relates to estimating fluid breakthrough times at producing well locations based on fluid propagation simulations.

BACKGROUND OF THE INVENTION

Various systems and methods are known for estimating the fluid breakthrough time at a producing well location, including history matching (HM). History Matching (HM) is a systematic procedure of altering a reservoir simulation model to reproduce the dynamic field response. In HM applications and conditioning reservoir models to production data, the main objectives are a) integration of production data into reservoir models; b) flexibility, cost-effectiveness and computational efficiency; and c) full utilization of dynamic data.

In the last decade, HM technology has evolved tremendously and gained major recognition and expansion from the traditional (i.e. manual, deterministic) approach, mostly built on stratigraphic methods to new developments like probabilistic, streamline-based HM, sensitivity/gradient-based and experimental design.

HM workflows largely consider the minimization of the misfit between the measured and simulated fluid (e.g. oil or water) dynamic response at the individual production well as one of the inversion main objectives. In water-flooding Enhanced Oil Recovery (EOR) studies, for example, the response misfit represents the differential or cumulative water-cut curves with two main attributes: 1) fluid breakthrough time; and 2) trend and shape of the response. While both attributes represent important variables in the process of misfit minimization, it is the fluid breakthrough time that bares the highest impact on the economics of the well production. Furthermore, the interval (i.e. time-frame) of the fluid breakthrough is always burdened with uncertainty, which makes the effort of estimation with highest confidence possible, even more relevant. In fact, it is a good practice in HM of dynamic well data to consider the breakthrough time as the first-order effect and the variations in curve trend/shape as the second-order effect, because they mainly reflect on the operating conditions.

Despite the progress in HM technology, it is still by far the most time-consuming aspect of the model building/simulation study and the HM workflow faces many difficulties, which include:

-   -   i) non-linear results between the production response and         reservoir parameters;     -   ii) non-unique solutions, which require a definition of some         semblance of “uniqueness”;     -   iii) the relative impact of key parameters may not be obvious;     -   iv) constraints are not bounded and uncertainties and in the         variables are seldom known; and     -   v) the production data may be noisy and inherently biased.

SUMMARY OF THE INVENTION

The present invention therefore, meets the above needs and overcomes one or more deficiencies in the prior art by providing systems and methods for estimating fluid breakthrough times at producing well locations based on fluid propagation simulations.

In one embodiment, the present invention includes a method for estimating a fluid breakthrough time at a production well based on fluid propagation simulation data, comprising: i) identifying streamline tracking data; ii) calculating an average streamline travel time in each grid-cell based on the streamline tracking data; iii) identifying a shortest or fastest streamline for the production well using the average streamline travel time in each grid-cell; iv) calculating an average time-of-flight for the shortest or fastest streamline over each traversed grid-cell using a computer processor; and v) estimating the fluid breakthrough time at the production well using the fluid propagation simulation data, and the average time-of-flight for the shortest or fastest streamline.

In another embodiment, the present invention includes a non-transitory program carrier device tangibly carrying computer executable instructions for estimating a fluid breakthrough time at a production well. The instructions being executable to implement: i) identifying streamline tracking data; ii) calculating an average streamline travel time in each grid-cell based on the streamline tracking data; iii) identifying a shortest or fastest streamline for the production well using the average streamline travel time in each grid-cell; iv) calculating an average time-of-flight for the shortest or fastest streamline over each traversed grid-cell using; and v) estimating the fluid breakthrough time at the production well using the fluid propagation simulation data, and the average time-of-flight for the shortest or fastest streamline.

Additional aspects, advantages and embodiments of the invention will become apparent to those skilled in the art from the following description of the various embodiments and related drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is described below with references to the accompanying drawings in which like elements are referenced with like reference numerals, and in which:

FIG. 1 is a flow diagram illustrating one embodiment of a method for implementing the present invention.

FIG. 2A illustrates the velocity and the direction of fluid propagating through a wide sand pocket.

FIG. 2B illustrates the velocity and the direction of fluid propagating through a narrow sand pocket.

FIG. 3 illustrates an example of fluid propagation through a sand fraction of a facies model during the initial stage of simulation.

FIG. 4A illustrates a synthetic 2D permeability model with 2500 grid-cells (50×50) and a 5-spot-pattern of wells (1 injection well (I) and 4 production wells (P₁-P₄)).

FIG. 4B illustrates a simulation of fluid propagation through the 2D permeability model in FIG. 4A from the injector well (I) in terms of the number of iterations (2500) that the simulation was run.

FIG. 5 illustrates a possible streamline distribution in the 5-spot-pattern of wells in FIG. 4B.

FIG. 6 illustrates the streamline travel time along its arc length within a given grid-cell (i,j,k) of a 2D permeability model.

FIG. 7A illustrates the observed (measured) water-cut curve for the producing well P₁ in FIG. 4A.

FIG. 7B illustrates the observed (measured) water-cut curve for the producing well P₂ in FIG. 4A.

FIG. 7C illustrates the observed (measured) water-cut curve for the producing well P₃ in FIG. 4A.

FIG. 7D illustrates the observed (measured) water-cut curve for the producing well P₄ in FIG. 4A.

FIG. 8 is a block diagram illustrating one embodiment of a system for implementing the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The subject matter of the present invention is described with specificity, however, the description itself is not intended to limit the scope of the invention. The subject matter thus, might also be embodied in other ways, to include different steps or combinations of steps similar to the ones described herein, in conjunction with other present or future technologies. Moreover, although the term “step” may be used herein to describe different elements of methods employed, the term should not be interpreted as implying any particular order among or between various steps herein disclosed unless otherwise expressly limited by the description to a particular order. While the present invention may be applied in the oil and gas industry, it is not limited thereto and may also be applied in other industries to achieve similar results.

The present invention includes systems and methods to estimate fluid breakthrough times at producing well locations based on the simulation of fluid propagation. The present invention includes a fluid propagation simulation, which is generally static and renders the invasion time(s) for fluid injected at the injection well(s) to reach the production well(s). The simulation affords full consideration of facies modeling, which preserves control over depositional continuity of geological models by directly constraining the simulation with the facies distribution. The simulation also preserves the stochasticity of the fluid front propogation. Despite the static nature of the simulation, the stochastic sampling of the moving fluid front is performed by using a uniform distribution.

The present invention converts the fluid invasion time(s) (given by the simulation in units of iterations) to the domain of physical time (given i.e. in days, weeks, months . . . ), which is compatible with the well production history. The present invention therefore, provides new possibilities for the rapid estimation of valuable well production parameters in a fast and cost-effective manner. For example, a fast and accurate estimation of the fluid breakthrough time(s) associated with an individual reservoir model can be achieved prior to commencing a full inversion. Such an estimate would provide valuable information to the well operators in terms of well valve dynamics, particularly in water/gas-flooding EOR projects where the management of oil and water/gas production bares a substantial economic impact.

In order to achieve rapid estimates of the fluid breakthrough time(s) (T^(BT)), the present invention uses the combination of streamline tracking and associated Time-Of-Flight (“TOF”) with the simulation. The present invention therefore, enables quick approximation of fluid breakthrough times following the simulation run and one iteration of streamline tracking in the process of streamline-sensitivity assisted Automated History Matching (“AHM”) of reservoir models.

Method Description

Referring now to FIG. 1, a flow diagram illustrates one embodiment of a method 100 for implementing the present invention.

In step 102, fluid propagation simulation (“FPS”) is performed. One technique for performing FPS is based on an algorithm in the RGeoS software package developed by D. Renard. The FPS algorithm simulates the distribution of several fluids known at the injection and/or production wells, which is conditioned by the facies information known at the nodes of a regular grid and tends to let the fluid encountered at the wells (e.g. the injection well) to grow or expand spatially. The velocity and the direction of growth depend on the size of the sand pockets that can be filled. In FIG. 2, for example, true velocity and the direction of fluid propagating through a wide sand pocket (FIG. 2A) and a narrow sand pocket (FIG. 2B) are illustrated. The larger the pocket 206, 208, the quicker the growth. Velocity vectors 202, 204 are utilized in the FPS algorithm. The FPS algorithm is designed to perform one simulation of a numerical variable using the Eden simulation technique. The technique provides a faster alternative solution for a multiphase fluid flow simulation program. The technique combines a dual medium “black and white” example where white represents sand and black represents shale with one or more injection wells and one or more production wells as illustrated in FIG. 3. In this example, the locations of sand facies 302, 304, 306, and of two injection wells 307, 308, are illustrated.

Referring now to FIG. 4A, a synthetic 2D permeability model is illustrated with 2500 grid cells (50×50) and a 5-spot-pattern of wells (1 injection well (I) and 4 production wells (P₁-P₄)). The FPS algorithm was executed in 2500 iterations because one cell of the model is populated per iteration. In FIG. 4B, a simulation of fluid propagation through the 2D permeability model in FIG. 4A from the injection well (I) is illustrated in terms of the number of iterations (2500) the simulation was run. In FIG. 5, one possible streamline distribution in the 5-spot-pattern of wells in FIG. 4B is illustrated.

In order to implement the FPS algorithm as a rapid proxy estimate of the fluid breakthrough time in AHM inversion of water-cut curves, the conversions of the fluid invasion time(s) to the domain of physical time(s) has to be considered with the following main assumptions:

-   -   i) streamline TOF represents a crucial normalization factor;     -   ii) tracking TOF from the production well(s) indicates the         drainage volume; and     -   iii) tracking fluid from the injection well gives an assessment         of swept volume.

For the estimation of fluid breakthrough time in a production well, it is assumed that the following computations are completed for a given reservoir model using any technique well known in the art to track streamlines based on a forward simulation of fluid pressure and velocity: a) the computation of fluid invasion time (i.e. step 102); and b) the first iteration of streamline tracking and TOF computation (i.e. step 106). These computations will provide a) the fluid invasion time from the FPS algorithm given by the number of simulation iterations (assuming 1 iteration per grid-cell); and b) the total number of streamlines traversing any reservoir model grid-cell with (i,j,k) coordinates.

In step 104, the FPS data results from step 102 are identified, which includes the fluid invasion time given by the number of simulation iterations needed for the fluid to reach any production well (P_(m)) from an injection well through one or more grid-cells representing the reservoir property model.

In step 106, the streamline tracking data are identified using any well known technique, which include the number of streamline segments traversing each grid-cell (N_(SLN)), the travel time (∂τ) for each streamline segment (ψ_(m,n) ^(i,j,k)) in each grid-cell, the grid-cell indices and the total number of grid-cells traversed by all streamlines connecting an injection well with a production well. Referring now to FIG. 6, the streamline travel time along its arc length within a given grid-cell of a 2D permeability model is illustrated. Indices (n) and (m) run over all streamline segments in each grid-cell and all production wells, respectively (n=[1 . . . N_(SLN)] and m=[1 . . . N_(P)]). Travel time ∂τ(ψ_(m,n) ^(i,j,k)) for a streamline segment in each grid-cell may be calculated by integrating the “slowness” of the streamline tracer along each streamline trajectory using the following equation:

$\begin{matrix} {{\partial{\tau \left( \psi_{m,n}^{i,j,k} \right)}} = {\int_{\psi}^{\;}{{\partial{s(x)}}\ {r}}}} & (1) \end{matrix}$

where ∂s(x) corresponds to the “slowness” of the streamline tracer (defined as the inverse of the tracer velocity) and dr corresponds to the arc length of the streamline segment (ψ_(m,n) ^(i,j,k)) between the inlet and outlet locations on the bounding surface of the grid-cell with (i,j,k) coordinates.

In step 108, the average streamline travel time in each grid-cell (∂{tilde over (τ)}) is calculated by taking into account all streamline segments traversing each grid-cell, which may be calculated using the following equation:

$\begin{matrix} {{\partial\overset{\sim}{\tau}} = {\frac{1}{N_{SLN}}{\sum\limits_{n = 1}^{N_{SLN}}{\partial{\tau \left( \psi_{m,n}^{i,j,k} \right)}}}}} & (2) \end{matrix}$

where (N_(SLN)) is the number of streamline segments traversing each grid-cell from step 106 and ∂τ(ψ_(m,n) ^(i,j,k)) is the travel time for each streamline segment in each grid-cell from step 106.

In step 114, the shortest/fastest streamline is identified for each production well (P_(m)) using the average streamline travel time in each grid-cell from step 108 and any well-known searching algorithm. The shortest/fastest streamline is the streamline with the lowest sum of average streamline travel times (∂{tilde over (τ)}^(min)) in the grid-cells the streamline traverses between an injection well (I) and a production well (P_(m)).

In step 116, the total number of all grid-cells ({circumflex over (N)}_(GC) ^(min)) traversed by the shortest/fastest streamline identified in step 114, and their indices from step 106, are stored.

In step 118, the average TOF (<TOF>^(min)) for the shortest/fastest streamline identified in step 114 is calculated over each traversed grid-cell using the lowest sum of average streamline travel times (∂{tilde over (τ)}^(min)) for the shortest/fastest streamline identified in step 114 and the total number of all grid-cells ({circumflex over (N)}_(GC) ^(min)) stored in step 116, which may be calculated using the following equation:

$\begin{matrix} {{\langle{TOF}\rangle}^{m\; i\; n} = {\frac{1}{{\hat{N}}_{GC}^{m\; i\; n}}{\sum\limits_{u = 1}^{{\hat{N}}_{GC}^{m\; i\; n}}{\partial{\overset{\sim}{\tau}}_{u}^{m\; i\; n}}}}} & (3) \end{matrix}$

where index (u) represents the number of runs over all indices of grid-cells traversed by the shortest/fastest streamline. The distinction between the “fastest” and the “slowest” streamline from the distribution of streamlines associated with each production well (P_(m)) is relevant to discriminate between the homogeneous and heterogeneous spatial distribution of reservoir properties such as, for example, channels. The difference between the distribution of streamlines in FIG. 5 reveals that production wells P₂ and P₃ are connected with injection well (I) through a distinctively different geological formation than production wells P₁ and P₄, which might correspond to an underlying channel structure.

In step 120, the method 100 determines if all grid-cells traversed by the shortest/fastest streamline have been considered. If all traversed grid-cells have not been considered, then the method 100 returns to step 118. If all traversed grid-cells have been considered, then the method 100 proceeds to step 124. Alternatively, steps 118 through 120 may be performed at the same time for each traversed grid-cell.

In step 124, an estimate of the fluid breakthrough time for each production well (P_(m)) is calculated by combining the streamline tracking data from step 106 with the FPS data from step 104, which may be calculated using the following equation:

$\begin{matrix} {T^{BT} = {{\langle{TOF}\rangle}^{m\; i\; n} \times \frac{t_{INV}^{i,j,k}}{N_{p}} \times \frac{N_{SLN}^{m}}{N_{xyz}}}} & (4) \end{matrix}$

where (N_(xyz)) and (N_(p)) represent the total size of the reservoir property model and the total number of production wells, respectively, (<TOF>^(min)) represents the average TOF for the shortest/fastest streamline calculated in step 118, (N_(SLN) ^(m)) represents the total number of grid-cells traversed by all streamlines connecting injection well (I) with a production well (P_(m)) and (t_(INV) ^(i,j,k)) represents the fluid invasion time from step 104.

In step 126, the method 100 determines if all production wells have been considered. If all production wells (P_(m)) have not been considered, then the method 100 returns to step 104. If all production wells (P_(m)) have been considered, then the method 100 ends. Alternatively, steps 104 through 126 may be performed at the same time for each production well (P_(m)).

Example

Referring now to the synthetic 2D permeability model in FIG. 4A, the observed (measured) water-cut curves for the configuration model in FIG. 4A are given in FIGS. 7A, 7B, 7C, and 7D for each of the four production wells (P₁, P₂, P₃, and P₄).

The date/time data points on the x-axis in FIGS. 7A-7D correspond with the physical dates associated with the water injection plan (water breakthrough data points) presented in Table 1 below:

TABLE 1 Data point Physical Date (dd/mm/yyyy) 1 17/9/2000 2 4/6/2001 3 19/2/2002 4 6/10/2002 5 24/7/2003 6 9/4/2004 7 25/12/2004 8 11/9/2005

The observed water breakthrough times deduced from FIG. 4A, are given in Table 2 below. Moreover, Table 2 lists the water invasion times calculated by the FPS algorithm the water breakthrough times (T^(B7)) calculated using the proposed method in FIG. 1 and the uncertainty associated with result obtained by the proposed method in FIG. 1.

TABLE 2 Invasion T^(BT) by Observed T^(BT) Time proposed Uncertainty Producer (days) (iterations) method (days) (days/%) P1 263 2272 281.41 18.41/+7% P2 121 2027 124.63 15.73/+3% P3 263 2268 239.33 −23.67/−9%  P4 1043 2491 1105.58 62.58/+6%

The results indicate that the proposed method in FIG. 1 is capable of rapidly predicting the fluid breakthrough time with an uncertainty of less than 10% for the given 5-spot-pattern of wells. The achieved uncertainty could be different (larger/smaller) when fluid propagation is applied through the field with significantly higher geological complexity and the dynamic model combines a significantly large number of producing wells.

System Description

The present invention may be implemented through a computer-executable program of instructions, such as program modules, generally referred to software applications or application programs executed by a computer. The software may include, for example, routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types. DecisionSpace® Desktop, which is a commercial software application marketed by Landmark Graphics Corporation, may be used as an interface application to implement the present invention. The software may also cooperate with other code segments to initiate a variety of tasks in response to data received in conjunction with the source of the received data. The software may be stored and/or carried on any variety of memory such as CD-ROM, magnetic disk, bubble memory and semiconductor memory (e.g., various types of RAM or ROM). Furthermore, the software and its results may be transmitted over a variety of carrier media such as optical fiber, metallic wire, and/or through any of a variety of networks, such as the Internet.

Moreover, those skilled in the art will appreciate that the invention may be practiced with a variety of computer-system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable-consumer electronics, minicomputers, mainframe computers, and the like. Any number of computer-systems and computer networks are acceptable for use with the present invention. The invention may be practiced in distributed-computing environments where tasks are performed by remote-processing devices that are linked through a communications network. In a distributed-computing environment, program modules may be located in both local and remote computer-storage media including memory storage devices. The present invention may therefore, be implemented in connection with various hardware, software, or a combination thereof, in a computer system or other processing system.

Referring now to FIG. 8, a block diagram illustrates one embodiment of a system for implementing the present invention on a computer. The system includes a computing unit, sometimes referred to as a computing system, which contains memory, application programs, a client interface, a video interface, and a processing unit. The computing unit is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention.

The memory primarily stores the application programs, which may also be described as program modules containing computer-executable instructions, executed by the computing unit for implementing the present invention described herein and illustrated in FIG. 2. The memory therefore, includes a fluid breakthrough time estimating module, which enables the methods illustrated and described in reference to FIG. 1 and integrates functionality from the remaining application programs illustrated in FIG. 8. The fluid breakthrough time estimating module, for example, may be used to execute many of the functions described in reference to the method 100 in FIG. 1. Decision Space® Desktop may be used, for example, as an interface application to implement the fluid breakthrough time estimating module and to utilize the results of the method 100 in FIG. 1.

Although the computing unit is shown as having a generalized memory, the computing unit typically includes a variety of computer readable media. By way of example, and not limitation, computer readable media may comprise computer storage media The computing system memory may include computer storage media in the form of volatile and/or nonvolatile memory such as a read only memory (ROM) and random access memory (RAM). A basic input/output system (BIOS), containing the basic routines that help to transfer information between elements within the computing unit, such as during start-up, is typically stored in ROM, The RAM typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by the processing unit. By way of example, and not limitation, the computing unit includes an operating system, application programs, other program modules, and program data.

The components shown in the memory may also be included in other removable/non-removable, volatile/nonvolatile computer storage media or they may be implemented in the computing unit through an application program interface (“API”) or cloud computing, which may reside on a separate computing unit connected through a computer system or network. For example only, a hard disk drive may read from or write to non-removable, nonvolatile magnetic media, a magnetic disk drive may read from or write to a removable, non-volatile magnetic disk, and an optical disk drive may read from or write to a removable, nonvolatile optical disk such as a CD ROM or other optical media. Other removable/non-removable, volatile/non-volatile computer storage media that can be used in the exemplary operating environment may include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The drives and their associated computer storage media discussed above provide storage of computer readable instructions, data structures, program modules and other data for the computing unit.

A client may enter commands and information into the computing unit through the client interface, which may be input devices such as a keyboard and pointing device, commonly referred to as a mouse, trackball or touch pad. Input devices may include a microphone, joystick, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit through a system bus, but may be connected by other interface and bus structures, such as a parallel port or a universal serial bus (USB).

A monitor or other type of display device may be connected to the system bus via an interface, such as a video interface. A graphical user interface (“GUI”) may also be used with the video interface to receive instructions from the client interface and transmit instructions to the processing unit. In addition to the monitor, computers may also include other peripheral output devices such as speakers and printer, which may be connected through an output peripheral interface.

Although many other internal components of the computing unit are not shown, those of ordinary skill in the art will appreciate that such components and their interconnection are well known.

While the present invention has been described in connection with presently preferred embodiments, it will be understood by those skilled in the art that it is not intended to limit the invention to those embodiments. It is therefore, contemplated that various alternative embodiments and modifications may be made to the disclosed embodiments without departing from the spirit and scope of the invention defined by the appended claims and equivalents thereof. 

1. A method for estimating a fluid breakthrough time at a production well based on fluid propagation simulation data, comprising: identifying streamline tracking data; calculating an average streamline travel time in each grid-cell based on the streamline tracking data; identifying a shortest or fastest streamline for the production well using the average streamline travel time in each grid-cell; calculating an average time-of-flight for the shortest or fastest streamline over each traversed grid-cell using a computer processor; and estimating the fluid breakthrough time at the production well using the fluid propagation simulation data and the average time-of-flight for the shortest or fastest streamline.
 2. The method of claim 1, wherein the fluid propagation simulation data comprises a fluid invasion time represented by a number of simulation iterations needed for a fluid to reach the production well from an injection well through one or more grid-cells representing a reservoir property model.
 3. The method of claim 1, wherein the streamline tracking data comprises a number of streamline segments traversing each grid-cell, a travel time for each streamline segment in each grid-cell, indices for each grid-cell and a total number of grid-cells traversed by all streamlines connecting an injection well with a production well.
 4. The method of claim 3, wherein the average streamline travel time in each grid-cell is calculated by: ${\partial\overset{\sim}{\tau}} = {\frac{1}{N_{SLN}}{\sum\limits_{n = 1}^{N_{SLN}}{\partial{\tau \left( \psi_{m,n}^{i,j,k} \right)}}}}$ wherein (N_(SLN)) represents the number of streamline segments traversing each (∂τ(ψ_(m,n) ^(i,j,k))) grid-cell and represents the travel time for each streamline segment in each grid-cell.
 5. The method of claim 1, wherein the shortest or fastest streamline for the production well represents a streamline with a lowest sum of average streamline travel times in grid-cells the streamline traverses between an injection well and the production well.
 6. The method of claim 5, wherein the average time-of-flight for the shortest or fastest streamline is calculated over each traversed grid-cell using the lowest sum of average streamline travel times for the shortest or fastest streamline and a total number of grid-cells traversed by the shortest or fastest streamline.
 7. The method of claim 6, wherein the average time-of-flight for the shortest or fastest streamline is calculated by: ${\langle{TOF}\rangle}^{m\; i\; n} = {\frac{1}{{\hat{N}}_{GC}^{m\; i\; n}}{\sum\limits_{u = 1}^{{\hat{N}}_{GC}^{m\; i\; n}}{\partial{\overset{\sim}{\tau}}_{u}^{m\; i\; n}}}}$ wherein ({circumflex over (N)}_(GC) ^(min)) represents the total number of all grid-cells traversed by the shortest or fastest streamline, (∂{tilde over (τ)}^(min)) represents the lowest sum of average streamline travel times for the shortest or fastest streamline and (u) represents a number of runs over all indices of grid-cells traversed by the shortest or fastest streamline.
 8. The method of claim 2, wherein the fluid breakthrough time at the production well is estimated by: $T^{BT} = {{\langle{TOF}\rangle}^{m\; i\; n} \times \frac{t_{INV}^{i,j,k}}{N_{p}} \times \frac{N_{SLN}^{m}}{N_{xyz}}}$ wherein (N_(xyz)) and (N_(p)) represent a total size of the reservoir property model and a total number of production wells, respectively, (<TOF>^(min)) represents the average time-of-flight for the shortest or fastest streamline, (N_(SLN) ^(m)) represents a total number of grid-cells traversed by all streamlines connecting an injection well with the production well and (t_(INV) ^(i,j,k)) represents the fluid invasion time.
 9. The method of claim 1, further comprising repeating the steps in claim 1 for each production well.
 10. The method of claim 1, wherein the reservoir property model is a permeability model.
 11. A non-transitory carrier device tangibly carrying computer executable instructions for estimating a fluid breakthrough time at a production well based on fluid propagation simulation data, the instructions being executable to implement: identifying streamline tracking data; calculating an average streamline travel time in each grid-cell based on the streamline tracking data; identifying a shortest or fastest streamline for the production well using the average streamline travel time in each grid-cell; calculating an average time-of-flight for the shortest or fastest streamline over each traversed grid-cell; and estimating the fluid breakthrough time at the production well using the fluid propagation simulation data and the average time-of-flight for the shortest or fastest streamline.
 12. The program carrier device of claim 11, wherein the fluid propagation simulation data comprises a fluid invasion time represented by a number of simulation iterations needed for a fluid to reach the production well from an injection well through one or more grid-cells representing a reservoir property model.
 13. The program carrier device of claim 11, wherein the streamline tracking data comprises a number of streamline segments traversing each grid-cell, a travel time for each streamline segment in each grid-cell, indices for each grid-cell and a total number of grid-cells traversed by all streamlines connecting an injection well with a production well.
 14. The program carrier device of claim 13, wherein the average streamline travel time in each grid-cell is calculated by: ${\partial\overset{\sim}{\tau}} = {\frac{1}{N_{SLN}}{\sum\limits_{n = 1}^{N_{SLN}}{\partial{\tau \left( \psi_{m,n}^{i,j,k} \right)}}}}$ wherein (N_(SLN)) is the number of streamline segments traversing each grid-cell and (∂τ(ψ_(m,n) ^(i,j,k))) represents the travel time for each streamline segment in each grid-cell.
 15. The program carrier device of claim 11, wherein the shortest or fastest streamline for the production well represents a streamline with a lowest sum of average streamline travel times in grid-cells the streamline traverses between an injection well and the production well.
 16. The program carrier device of claim 15, wherein the average time-of-flight for the shortest or fastest streamline is calculated over each traversed grid-cell using the lowest sum of average streamline travel times for the shortest or fastest streamline and a total number of grid-cells traversed by the shortest or fastest streamline.
 17. The program carrier device of claim 16, wherein the average time-of-flight for the shortest or fastest streamline is calculated by: ${\langle{TOF}\rangle}^{m\; i\; n} = {\frac{1}{{\hat{N}}_{GC}^{m\; i\; n}}{\sum\limits_{u = 1}^{{\hat{N}}_{GC}^{m\; i\; n}}{\partial{\overset{\sim}{\tau}}_{u}^{m\; i\; n}}}}$ wherein ({circumflex over (N)}_(GC) ^(min)) represents the total number of all grid-cells traversed by the shortest or fastest streamline, (∂{tilde over (τ)}^(min)) represents the lowest sum of average streamline travel times for the shortest or fastest streamline and (u) represents a number of runs over all indices of grid-cells traversed by the shortest or fastest streamline.
 18. The program carrier device of claim 12, wherein the fluid breakthrough time at the production well is estimated by: $T^{BT} = {{\langle{TOF}\rangle}^{m\; i\; n} \times \frac{t_{INV}^{i,j,k}}{N_{p}} \times \frac{N_{SLN}^{m}}{N_{xyz}}}$ wherein (N_(xyz)) and (N_(p)) represent a total size of the reservoir property model and a total number of production wells, respectively, (<TOF>^(min)) represents the average time-of-flight for the shortest or fastest streamline, (N_(SLN) ^(m)) represents a total number of grid-cells traversed by all streamlines connecting an injection well with the production well and t_(INV) ^(i,j,k) represents the fluid invasion time.
 19. The program carrier device of claim 11, further comprising repeating the steps in claim 1 for each production well.
 20. The program carrier device of claim 11, wherein the reservoir property model is a permeability model. 